Finite Group Databasebeta

← #13173082

Subgroups — #13173082

Order 1728.

1582 conjugacy classes of subgroups; only the 275 normal subgroups are shown.

normal subgroups only

This group has 1582 conjugacy classes of subgroups — too many to draw. Only its 275 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×C32×A4C211yesmaximal, characteristic
2C2×C3⋊(C3⋊SL(2,ℤ/4))C212yesmaximal, complemented
3#10593666—312nomaximal
4#10592946—31nomaximal
5C22×C32×A4—22no—
6C22×C32×A4C412yescomplemented
7C22×C32×A4C2211yescharacteristic
8C3⋊(C3⋊SL(2,ℤ/4))—22no—
9#88864—41nomaximal
10C25×C32D311yescharacteristic
11C23×C3×A4D3112yes—
12C2×C3⋊SL(2,ℤ/4)—324no—
13C2×#3757—32no—
14C2×#3757—31no—
15C2×#3757—31no—
16C2×#3757—32no—
17C23×C33—41no—
18C2×C32×A4—22no—
19C2×C32×A4C2×C411yescharacteristic
20C2×C32×A4D412yes—
21C2×C3⋊(C3⋊Dic3)—42no—
22#5432—99no—
23#5247—94no—
24C24×C32—22no—
25C24×C32—31no—
26C24×C32—31no—
27C24×C32—32no—
28C24×C32—62no—
29C24×C32Dic312yescomplemented
30C24×C32C2×D311yescharacteristic
31C22×C3×A4—224no—
32C22×C3×A4Dic3124yescomplemented
33C22×C3×A4C2×D3112yes—
34C22×C3⋊Dic3—32no—
35C22×C3⋊Dic3—32no—
36C3⋊SL(2,ℤ/4)—624no—
37#3757—62no—
38#3757—64no—
39#3757—62no—
40#3757—61no—
41#3757—62no—
42#3757—61no—
43#3757—1212no—
44C22×C33—41no—
45C22×C33—42no—
46C22×C33—82no—
47C32×A4C22⋊C411yescharacteristic, complemented
48C3⋊(C3⋊Dic3)—82no—
49C25×C3C3⋊D314yes—
50C23×A4C3⋊D319yes—
51C2×SL(2,ℤ/4)—918no—
52C2×#217—98no—
53C2×#217—94no—
54C2×#217—94no—
55C2×#217—98no—
56C23×C32—22no—
57C23×C32—31no—
58C23×C32—32no—
59C23×C32—31no—
60C23×C32—32no—
61C23×C32—34no—
62C23×C32—32no—
63C23×C32—32no—
64C23×C32—31no—
65C23×C32—412no—
66C23×C32—62no—
67C23×C32—62no—
68C23×C32—62no—
69C23×C32—61no—
70C23×C32—64no—
71C23×C32—62no—
72C23×C32—64no—
73C23×C32C2×Dic311yescharacteristic
74C23×C32#6712yes—
75C23×C32S411yescharacteristic
76C2×C3×A4—224no—
77C2×C3×A4C2×Dic3112yes—
78C2×C3×A4#67124yes—
79C2×C3⋊Dic3—62no—
80C2×C3⋊Dic3—62no—
81C2×C3⋊Dic3—62no—
82C2×C3⋊Dic3—62no—
83C2×C3⋊Dic3—62no—
84C2×C3⋊Dic3—62no—
85C2×C3⋊Dic3—1224no—
86#379—271noSylow
87C2×C33—41no—
88C2×C33—42no—
89C2×C33—82no—
90C24×C3—28no—
91C24×C3—34no—
92C24×C3—34no—
93C24×C3—38no—
94C24×C3—68no—
95C24×C3C3⋊Dic318yescomplemented
96C24×C3C2×C3⋊D314yes—
97C22×A4—218no—
98C22×A4C3⋊Dic3118yescomplemented
99C22×A4C2×C3⋊D319yes—
100C22×Dic3—98no—
101C22×Dic3—98no—
102SL(2,ℤ/4)—1818no—
103#217—188no—
104#217—1816no—
105#217—188no—
106#217—184no—
107#217—188no—
108#217—184no—
109#217—369no—
110C22×C32—22no—
111C22×C32—31no—
112C22×C32—32no—
113C22×C32—31no—
114C22×C32—31no—
115C22×C32—32no—
116C22×C32—32no—
117C22×C32—34no—
118C22×C32—32no—
119C22×C32—412no—
120C22×C32—424no—
121C22×C32—62no—
122C22×C32—64no—
123C22×C32—62no—
124C22×C32—61no—
125C22×C32—62no—
126C22×C32—64no—
127C22×C32—62no—
128C22×C32—824no—
129C22×C32#21711yescharacteristic, complemented
130C22×C32SL(2,ℤ/4)12yescomplemented
131C22×C32C2×S411yescharacteristic
132C3×A4#217112yescomplemented
133C3⋊Dic3—122no—
134C3⋊Dic3—122no—
135C3⋊Dic3—2424no—
136C25C3⋊(C3⋊D3)11yescharacteristic
137C2×C22⋊C4—272no—
138C2×C22⋊C4—271no—
139C2×C22⋊C4—271no—
140C2×C22⋊C4—272no—
141C33—41noSylow
142C23×C3—28no—
143C23×C3—34no—
144C23×C3—38no—
145C23×C3—38no—
146C23×C3—34no—
147C23×C3—316no—
148C23×C3—34no—
149C23×C3—38no—
150C23×C3—38no—
151C23×C3—49no—
152C23×C3—68no—
153C23×C3—68no—
154C23×C3—68no—
155C23×C3—64no—
156C23×C3—616no—
157C23×C3—68no—
158C23×C3—616no—
159C23×C3C2×C3⋊Dic314yes—
160C23×C3#63818yes—
161C23×C3C3⋊S414yes—
162C2×A4—218no—
163C2×A4C2×C3⋊Dic319yes—
164C2×A4#638118yes—
165C2×Dic3—188no—
166C2×Dic3—188no—
167C2×Dic3—188no—
168C2×Dic3—188no—
169C2×Dic3—188no—
170C2×Dic3—188no—
171C2×Dic3—3618no—
172C2×C32—22no—
173C2×C32—32no—
174C2×C32—31no—
175C2×C32—31no—
176C2×C32—412no—
177C2×C32—424no—
178C2×C32—62no—
179C2×C32—824no—
180C2×C32C2×SL(2,ℤ/4)11yescharacteristic
181C2×C32GL(2,ℤ/4)12yes—
182C24—22no—
183C24—31no—
184C24—31no—
185C24—32no—
186C24—62no—
187C24C3⋊(C3⋊Dic3)12yescomplemented
188C24C2×C3⋊(C3⋊D3)11yescharacteristic
189C22×C4—272no—
190C22×C4—272no—
191C22⋊C4—542no—
192C22⋊C4—544no—
193C22⋊C4—542no—
194C22⋊C4—541no—
195C22⋊C4—542no—
196C22⋊C4—541no—
197C22×C3—28no—
198C22×C3—34no—
199C22×C3—38no—
200C22×C3—34no—
201C22×C3—34no—
202C22×C3—38no—
203C22×C3—38no—
204C22×C3—316no—
205C22×C3—38no—
206C22×C3—49no—
207C22×C3—418no—
208C22×C3—68no—
209C22×C3—616no—
210C22×C3—68no—
211C22×C3—64no—
212C22×C3—68no—
213C22×C3—616no—
214C22×C3—68no—
215C22×C3—818no—
216C22×C3#375714yescomplemented
217C22×C3C3⋊SL(2,ℤ/4)18yescomplemented
218C22×C3C2×C3⋊S414yes—
219A4#375719yescomplemented
220Dic3—368no—
221Dic3—368no—
222Dic3—7218no—
223C32—412no—
224C32#543211yescharacteristic, complemented
225C23—22no—
226C23—31no—
227C23—32no—
228C23—32no—
229C23—31no—
230C23—34no—
231C23—31no—
232C23—32no—
233C23—32no—
234C23—62no—
235C23—62no—
236C23—62no—
237C23—61no—
238C23—64no—
239C23—62no—
240C23—64no—
241C23C2×C3⋊(C3⋊Dic3)11yescharacteristic
242C23#631212yes—
243C23C3⋊(C3⋊S4)11yescharacteristic
244C2×C4—542no—
245C2×C4—542no—
246C2×C4—542no—
247C2×C4—542no—
248C2×C4—542no—
249C2×C4—542no—
250C2×C3—28no—
251C2×C3—38no—
252C2×C3—34no—
253C2×C3—34no—
254C2×C3—49no—
255C2×C3—418no—
256C2×C3—68no—
257C2×C3—818no—
258C2×C3C2×C3⋊SL(2,ℤ/4)14yes—
259C2×C3#6426918yes—
260C22—22no—
261C22—31no—
262C22—32no—
263C22—31no—
264C22—31no—
265C22—32no—
266C22—32no—
267C22—34no—
268C22—32no—
269C22—62no—
270C22—64no—
271C22—62no—
272C22—61no—
273C22—62no—
274C22—64no—
275C22—62no—
276C22#8886411yescharacteristic, complemented
277C22C3⋊(C3⋊SL(2,ℤ/4))12yescomplemented
278C22C2×C3⋊(C3⋊S4)11yescharacteristic
279C4—1082no—
280C4—1082no—
281C3—49no—
282C3#1059366614yescomplemented
283C2—22no—
284C2—32no—
285C2—31no—
286C2—31no—
287C2—62no—
288C2C2×C3⋊(C3⋊SL(2,ℤ/4))11yescharacteristic
289C2#1172447312yes—
subgroupG/H
center, FrattiniC22C2×C3⋊(C3⋊S4)
commutatorC2×C32×A4C2×C4
FittingC25×C32D3
socleC24×C32C2×D3
prime radicalC23×C32×A4C2
cosocleC22×C32×A4C22
hypercenterC23C3⋊(C3⋊S4)
hypocenterC32×A4C22⋊C4
LCSG ▷ C2×C32×A4 ▷ C32×A4
UCSC1 ◁ C22 ◁ C23

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. C2×C3⋊(C3⋊SL(2,ℤ/4))C2semidirect
6. C22×C32×A4C4semidirect
29. C24×C32Dic3semidirect
32. C22×C3×A4Dic3semidirect
47. C32×A4C22⋊C4semidirect
95. C24×C3C3⋊Dic3semidirect
98. C22×A4C3⋊Dic3semidirect
129. C22×C32#217semidirect
130. C22×C32SL(2,ℤ/4)semidirect
132. C3×A4#217semidirect
187. C24C3⋊(C3⋊Dic3)semidirect
216. C22×C3#3757semidirect
217. C22×C3C3⋊SL(2,ℤ/4)semidirect
219. A4#3757semidirect
224. C32#5432semidirect
276. C22#88864semidirect
277. C22C3⋊(C3⋊SL(2,ℤ/4))semidirect
282. C3#10593666semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C23×C32×A42
C2×C3⋊(C3⋊SL(2,ℤ/4))2
#105936663
#105929463
#888644